The following post was written by Belinda Paine, an eighth grade special education teacher and one of my credential students. She wrote this string and used it both with her eighth grade students and her fellow credential students. Enjoy!
As an instructor of students with disabilities, I review many concepts in math that my students find challenging. One concept is the idea and notion of decimals. At first I thought students just visually couldn’t see the decimal, then I realized they were overwhelmed by the idea of place value. My students seemed to ignore the decimal altogether or completely gave up on the problem. I wrote this number string to help my students connect decimals to strategies we used for whole numbers, like compensation.
14.6 + .4
14.3 + .7
14.3 + .9
7.99 + .01
7.99 + .03
7.81 + .19
2.76 + 1.25
.12 + 4.89
I thought that negative reaction to decimals would only be my eight grade students. Little did I know, that when presenting this same number string on decimals to adults, the other students in my credential class, I would get the same reaction. Hands on the forehead, pressing their strands of hair back, eyes wide open and the dreaded sounds of, “Ughhhhhhh” from my peers. It blew me away. Many are uncomfortable with dealing with decimals.
I have now taught this number string to both adults and my students. This reflection moves back and forth between both teaching experiences. Before I started the string, I reminded my students that mathematicians could use concepts of money to help them understand the problems. My storyline: “I went to Target this weekend, and I had 14.40 cents in my wallet. Lucky me, before I got to the doors, I noticed .40 cents on the ground. How much could I spend in Target if I could only spend this cash?”
As a see thumbs go up, I noticed some students had already given up at the first sight of a decimal. I asked a student who looked confident to answer the question.
I smile in approval and respond with their favorite, “Now, tell me how you got that answer..” The student says confidently, “Well I knew that if I just added .60 to .40, I get $1.00, So I could just add 1.00 to 14.00 and get 15.00 altogether.”
“Could we represent that on a number line?”
“Yes, start with the cents, then at the dollars at the end.”
( a reinforcing problem of friendly numbers, “get to ten”- students who once gave up are using strategies that the student before just explained)
“15.2” an adult says. “Could you explain your thinking?” I ask. “yeah, well I just took the .9 and added .1 to get 1.00. I added the 1.00 to the now 14.2.”
“Awesome! This strategy is compensation. Using the idea/concept of tens to help solve the scary decimal problem! Let’s see it on a number line.”
For my students I would remind them how many pennies or cents are in a dollar before beginning this problem (some have not memorized this as a known fact). Adults initially seemed confused, but soon realized this was something they had memorized as a known fact. Thumbs quickly shot up in confidence.
They say, and I hear mumbling of “oh that’s so easy…” The student explains, “I know that 100 cents are in a 1.00, so .99 cents plus .01 cent is a 1.00. SO it’s just 7.00+1.00, which is 8.00.” “Great, perfect thinking, we will show that on a number line.”
7.99 + .03
“I think it’s uhhh 8.2” “Ok, great, can you tell me how you got that?” “I just did it like the problem before, where I look .99 and added a penny. Then I got the dollar and added it to the 7, which is 8. Then I had the 2 left over for the decimal, so it’s 8.2” Thumbs fly up.
“I got a different answer, but solved it almost the same way. I did that c word- compensation, where I added and took away a penny at the same time to get $8 but then it’s only two pennies, which is shown like point 02, which is 8.02, not 8.20.” “Great, thank you for that, does everyone see the difference between two pennies and twenty cents? We can look on the number line.”
7.81 + .19
(helper problem similar to the problem above that helps with place value of hundreths- adults and students used friendly numbers- making .19, .20, then adding it to .80 to get a dollar, left with a penny to subtract, getting 8.00 or compensation)
2.76 + 1.25
(I made sure students remembered how many quarters are in a dollar, and how that could possibly help them with this problem.)
4.01 “I knew that there were 4 quarters in a dollar. I can split up .76 to .75+.01. And .75 is three quarters, so I could just add that quarter to .75 to get 1.00. So I have 2.00+1.00+1.00, which is 4.00. The I had that extra penny, which makes the total 4.01.” “Wonderful! Does everyone see what she did there? Could we represent this on a number line?”
.12 + 4.89
It’s interesting how the placement of certain numbers can mix up a person’s thinking. Initially, all the problems had been shown with the bigger number first, and then the smaller number added to it. I saw big eyes for a second, but the students quickly realized there was an easier way to solve the problem without using an algorithm. It was not a scary decimal problem! Students and adults both solved this problem by making friendly numbers or using compensation. Some thinkers used +/-.11 using compensation to make the friendly 1.00, add it to the 4, and have a penny left over to make 5.01. And it was great to see students who were shutting down now attempting to solve the problem mentally, breaking the problem up into numbers they knew. Some added .89+.01 to make .90 then added .11 to make 1.01, next adding the 4.00. The grand total still being 5.01.
Students and adults proceeded to tell me that they thought, “it was going to be harder.” I even heard, “My head still hurts, but now the numbers don’t seem so scary.” As an instructor who accepts “the struggle” as learning, I appreciated these comments, hoping they wouldn’t be daunted by the idea of decimals in the real world.